Variance Gamma Model Research Articles

The logarithmic Schrödinger operator and associated Dirichlet problems

In this note, we study the integrodifferential operator (I−Δ)log corresponding to the logarithmic symbol log⁡(1+|ξ|2), which is a singular integral operator given by(I−Δ)logu(x)=dN∫RNu(x)−u(x+y)|y|Nω(|y|)dy, where dN=π−N2, ω(r)=21−N2rN2KN2(r) and Kν is the modified Bessel function of second kind with index ν. This operator is the Lévy generator of the variance gamma process and arises as derivative ∂s|s=0(I−Δ)s of fractional relativistic Schrödinger operators at s=0. In order to study associated Dirichlet problems in bounded domains, we first introduce the functional analytic framework and some properties related to (I−Δ)log, which allow to characterize the induced eigenvalue problem and Faber-Krahn type inequality. We also derive a decay estimate in RN of the Poisson problem and investigate small order asymptotics s→0+ of Dirichlet eigenvalues and eigenfunctions of (I−Δ)s in a bounded open Lipschitz set.

VARIANCE GAMMA MODEL AND ITS DEVELOPMENT FOR STOCKS CALL OPTION PRICES ESTIMATION

One of the developments in the options market is the formation of various pricing models for an option to help the buyer determine the fairness of the price. Black-Scholes model uses the assumption that the log return price of stocks is normally distributed, while in reality, the real-world price couldn’t fit into that assumption. To be able to obtain an option price calculation that considers skewness and kurtosis in the stock price data, there are many alternative methods, namely Black-Scholes with Gram-Charlier expansion and Variance Gamma models. As a development of the Variance Gamma model, there are also several methods that are able to reduce the simulations variance generated by the Variance Gamma model, namely Antithetic Variate Variance Gamma model and the Importance Sampling Variance Gamma method. For the results, the Importance Sampling Variance Gamma model and the Antithetic Variate Variance Gamma model are really able to reduce the resulting simulations variance so that it can produce more accurate option prices with market prices compared to the Variance Gamma model. In the end, all Variance Gamma models are able to produce a call option price that is more in line with the market price than all Black Scholes models.

Robust and Nearly Exact Option Pricing with Bilateral Gamma Processes

Bilateral gamma processes generalize the variance gamma process and allow one to capture, more precisely, the differences between upward and downward moves of financial returns, notably in terms of jump speed, frequency, and size. Like in most other pure jump models, option pricing under bilateral gamma processes relies heavily on numerical evaluation of Fourier integrals. In this article, we combine the Mellin transform and residue calculus to establish closed-form pricing formulas for several vanilla and exotic European options. These formulas take the form of series whose terms are straightforward to evaluate in practice and achieve an arbitrary degree of precision, without requiring sophisticated numerical tools; moreover, the convergence of the series is particularly accelerated for short maturity options, which are the most challenging to price for competing Fourier methods. Accuracy of the formulas is assessed thanks to several comparisons with state-of-the-art Fourier methods, with reference prices provided for future research.

Quasi-Monte Carlo simulation for American option sensitivities

The estimation of American option sensitivities is a challenging problem in financial engineering due to the stopping time involved. Progress has been made in using Monte Carlo (MC) simulation to obtain the estimation. However, the applicability of quasi-Monte Carlo (QMC) methods on estimating American option sensitivities is open. In this paper, we address this problem and propose efficient QMC methods for estimating American option sensitivities. Dimension reduction techniques, such as the Brownian bridge and principal component analysis, are used for further efficiency improvements. Numerical experiments in the cases of single underlying asset and multiple underlying assets under the Black–Scholes model and the variance gamma model demonstrate that QMC with dimension reduction techniques can significantly reduce the variance of the estimators of American option sensitivities.

Variance Gamma process as degradation model for prognosis and imperfect maintenance of centrifugal pumps

This paper considers the degradation modelling of a non-monotonous health indicator. A Variance Gamma process is proposed for the degradation modelling and its calibration in presence of data is discussed. The remaining useful lifetime estimation based on this model is considered and its sensitivity to parameters estimates is analysed. The model is applied to real data of leakage rate of centrifugal pump. Eventually, an imperfect maintenance policy is proposed and optimized.

VARIANCE GAMMA PROCESS WITH MONTE CARLO SIMULATION AND CLOSED FORM APPROACH FOR EUROPEAN CALL OPTION PRICE DETERMINATION

The Option is widely applied in the financial sector. The Black-Scholes-Merton model is often used in calculating option prices on a stock price movement. The model uses geometric Brownian motion which assumes that the data is normally distributed. However, in reality, stock price movements can cause sharp spikes in data, resulting in nonnormal data distribution. So we need a stock price model that is not normally distributed. One of the fastest growing stock price models today is the process exponential model. The process has the ability to model data that has excess kurtosis and a longer tail (heavy tail) compared to the normal distribution. One of the members of the process is the Variance Gamma (VG) process. The VG process has three parameters which each of them, to control volatility, kurtosis and skewness. In this research, the secondary data samples of options and stocks of two companies were used, namely zoom video communications, Inc. (ZM) and Nokia Corporation (NOK). The price of call options is determined by using closed form equations and Monte Carlo simulation. The Simulation was carried out for various values until convergent result was obtained.

The Risk Measurement under the Variance-Gamma Process with Drift Switching

The paper discusses an extension of the variance-gamma process with stochastic linear drift coefficient. It is assumed that the linear drift coefficient may switch to a different value at the exponentially distributed time. The size of the drift jump is supposed to have a multinomial distribution. We have obtained the distribution function, the probability density function and the lower partial expectation for the considered process in closed forms. The results are applied to the calculation of the value at risk and the expected shortfall of the investment portfolio in the related multivariate stochastic model.

The Variance Gamma++ process and applications to energy markets

AbstractThe purpose of this article is to introduce a new Lévy process, termed the Variance Gamma++ process, to model the dynamics of assets in illiquid markets. Such a process has the mathematical tractability of the Variance Gamma process and is obtained by applying the self‐decomposability of the gamma law. Compared to the Variance Gamma model, it has an additional parameter representing the measure of the trading activity. We give a full characterization of the Variance Gamma++ process in terms of its characteristic triplet, characteristic function, and transition density. In addition, we provide efficient path simulation algorithms, both forward and backward in time. We also obtain an efficient “integral‐free” explicit pricing formula for European options. These results are instrumental to apply Fourier‐based option pricing and maximum likelihood techniques for the parameter estimation. Finally, we apply our model to illiquid markets, namely to the calibration of European power futures market data. We accordingly evaluate exotic derivatives using the Monte Carlo method and compare these values to those obtained using the Variance Gamma process and give an economic interpretation. Finally, we illustrate an extension to the multivariate framework.

Vasicek interest rate model under Lévy process and pricing bond option

In this paper, we concentrate on the European option pricing problem in incomplete markets by taking Vasicek stochastic interest rate model into account and expanding it in accordance with a Lévy process named Variance Gamma (VG) process. In doing so, first, we prove the existence and uniqueness of the solution to the stochastic differential equation of this model. Then, by the means of the hedging and replicating techniques, pricing formulas for the bond and, in succession, for European options are established. Finally, from a numerical point of view, we will analyze the value of a bond based on our consideration model and also, we will study a European call option bond price. All these results are reported through the Monte-Carlo simulation method.

Calibration for multivariate Lévy-driven Ornstein-Uhlenbeck processes with applications to weak subordination

Consider a multivariate Lévy-driven Ornstein-Uhlenbeck process where the stationary distribution or background driving Lévy process is from a parametric family. We derive the likelihood function assuming that the innovation term is absolutely continuous. Two examples are studied in detail: the process where the stationary distribution or background driving Lévy process is given by a weak variance alpha-gamma process, which is a multivariate generalisation of the variance gamma process created using weak subordination. In the former case, we give an explicit representation of the background driving Lévy process, leading to an innovation term which is a discrete and continuous mixture, allowing for the exact simulation of the process, and a separate likelihood function. In the latter case, we show the innovation term is absolutely continuous. The results of a simulation study demonstrate that maximum likelihood numerically computed using Fourier inversion can be applied to accurately estimate the parameters in both cases.

Gamma variance model: Fractional Fourier Transform (FRFT)

The paper examines the Fractional Fourier Transform (FRFT) based technique as a tool for obtaining the probability density function and its derivatives; and mainly for fitting stochastic model with the fundamental probabilistic relationships of infinite divisibility. The probability density functions are computed, and the distributional proprieties are reviewed for Variance-Gamma (VG) model. The VG model has been increasingly used as an alternative to the Classical Lognormal Model (CLM) in modelling asset prices. The VG model was estimated by the FRFT. The data comes from the SPY ETF historical prices. The Kolmogorov-Smirnov (KS) goodness-of-fit shows that the VG model fits better the cumulative distribution of the sample data than the CLM. The best VG model comes from the FRFT estimation.

Daily stock prices prediction using variance gamma model

The Black-Scholes-Merton model is often used for modeling company assets, which requires normally distributed data. The company's asset price fluctuates greatly, causing the form of data distribution to allow for heavy tails, asymmetry and excess kurtosis. Therefore we need a model that can capture this phenomenon. The model is the Variance Gamma model, abbreviated as VG. The assumption of normality in the Black-Sholes-Merton theory is unable to capture the heavy tail and the asymmetry that exists in the log returns asset. This form of density usually that is too high compared to normal density is known as excess kurtosis. The additional parameters in the VG process can overcome heavy tail, asymmetry and excess kurtosis. These parameters can control the kurtosis, asymmetry, and adjust the slope of the log asset density. The VG process can be obtained from two approaches. First, the VG process as a change in Brownian motion to Gamma time, abbreviated as VG1. Second, the VG process is obtained by the difference of the two Gamma processes, abbreviated as VG2. In this study we applied the VG model to predict the daily stock price of PT Bank Negara Indonesia (Persero) Tbk. The results obtained show that the VG model provides accurate daily stock price prediction results.

No country for old distributions? On the comparison of implied option parameters between the Brownian motion and variance gamma process

Advanced stochastic approaches are often suggested as a solution to real-world derivative pricing inconsistencies like the non-linearity of the implied volatility smile. Using a novel high-frequency data set with over one million option trades and corresponding order books from the German market, we compare the normal distribution approach with a variance gamma process, which is – as a pure jump process – especially suitable for tick-by-tick data. We are able to report a flattened implied volatility smile with the variance gamma process. Other low-frequency results like time, information, and underlying moment dependencies for both stochastic processes are unchanged. All in-sample residuals of the normal distribution have a smaller variance below the 1% significance level compared to the variance gamma process. Additionally, we reveal a mean-reversion process. We show that the normal distribution is superior to the variance gamma process in an out-of-sample context and conclude that – even in a high-frequency environment – it is still “a country for old distributions”.

Option Pricing with Greed and Fear Factor: The Rational Finance Approach

In this article, we explain main concepts of prospect theory and cumulative prospect theory within the rational dynamic asset pricing framework. We derive option pricing formulas when asset returns are altered by a generalized prospect theory value function or a modified Prelec’s weighting probability function. We introduce new parametric classes for prospect theory value functions and probability weighting functions consistent with rational dynamic pricing theory. After studying the behavioral finance notion of “greed and fear” from the perspective of rational dynamic asset pricing theory, we derive the corresponding option pricing formulas when asset returns follow continuous diffusions or discrete binomial trees. We define a mixed subordinated variance gamma process to model asset return and derive the corresponding option pricing formula. Finally, we apply the proposed probability weighting functions to study the greedy or fearful disposition of option traders when asset returns follow a mixed subordinated variance gamma process. The results indicate availability bias and diminishing sensitivity of option traders.

Default and prepayment options pricing and default probability valuation under VG model

In this paper, a new approach, the Variance Gamma (VG) model, which is used to capture unexpected shocks (e.g., Covid-19) in housing markets, is proposed to contribute to the standard option-based mortgage valuation methods. Based on the VG model, the closed-form solutions are performed for pricing mortgage default and prepayment options. It solves the options pricing equations explicitly and illustrates numerical results for both mortgage default and prepayment options’ prices. Furthermore, the study enables researchers to monitor the default probability of mortgagors. Analyzing the effect of risks on default and prepayment options using simulations shows that the VG model captures the systematic and systemic (idiosyncratic) risks of default and prepayment options prices with closed-form solutions and computes the mortgage default probabilities. Therefore, it allows lenders a more advanced decision process compared to the standard option-based mortgage valuation method.

Variance gamma for stock model performance with excess kurtosis

The Geometric Brownian Motion (GBM) model is used widely for model the dynamic of asset price movement. One of the company’s assets is a stock. The distribution of stock data that is normally distributed can be modeled using the Geometric Brownian Motion model. However, the distribution of stock data showed excess kurtosis and tail when using the Brown Geometry Motion model was less precise. One model for data showing excess kurtosis and tail was Variance Gamma (VG). In this research, the sample used was the stock data of PT Bank Danamon Indonesia Tbk for the period April 25 th , 2018 to April 24 th , 2020. The data sample was divided into two parts, namely training data and testing data. Based on the result of the stock description statistics, the value of skewness = -2.105417 and kurtosis = 22.16438 was obtained, while hypothesis testing concluded that the stock distribution did not spread normally. The resulting parameters for the VG model were σ = 0.08071, v = 8.00500 and θ = 0.01976. Based on the results of testing on the last 38 observations, the MAPE value was = 6.97560%. These results gave the conclusion that the VG model provided excellent forecasting results.

Residue Sum Formula for Pricing Options under the Variance Gamma Model

We present and prove a triple sum series formula for the European call option price in a market model where the underlying asset price is driven by a Variance Gamma process. In order to obtain this formula, we present some concepts and properties of multidimensional complex analysis, with particular emphasis on the multidimensional Jordan Lemma and the application of residue calculus to a Mellin–Barnes integral representation in C3, for the call option price. Moreover, we derive triple sum series formulas for some of the Greeks associated to the call option and we discuss the numerical accuracy and convergence of the main pricing formula.

Prognostic and Classification of Dynamic Degradation in a Mechanical System Using Variance Gamma Process

Recently, maintaining a complex mechanical system at the appropriate times is considered a significant task for reliability engineers and researchers. Moreover, the development of advanced mechanical systems and the dynamics of the operating environments raises the complexity of a system’s degradation behaviour. In this aspect, an efficient maintenance policy is of great importance, and a better modelling of the operating system’s degradation is essential. In this study, the non-monotonic degradation of a centrifugal pump system operating in the dynamic environment is considered and modelled using variance gamma stochastic process. The covariates are introduced to present the dynamic environmental effects and are modelled using a finite state Markov chain. The degradation of the system in the presence of covariates is modelled and prognostic results are analysed. Two machine learning algorithms k-nearest-neighbour (KNN) and neural network (NN) are applied to identify the various characteristics of degradation and the environmental conditions. A predefined degradation threshold is assigned and used to propose a prognostic result for each classification state. It was observed that this methodology shows promising prognostic results.

Fast pricing of American options under variance gamma

We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference method and the simulation method can be used for pricing under this model, but their speed is not satisfactory. So there is a need for fast but accurate approximation methods. In the case of Black-Merton-Scholes model, there are fast approximation methods, but they cannot be utilized for the variance gamma model. We develop a new fast method inspired by the quadratic approximation method, while reducing the error by making use of a machine learning technique on pre-calculated quantities. We compare the performance of our proposed method with those of the existing methods and show that this method is efficient and accurate for practical use.

Daily stock prices prediction using variance gamma model

The Black-Scholes-Merton model is often used for modeling company assets, which requires normally distributed data. The company's asset price fluctuates greatly, causing the form of data distribution to allow for heavy tails, asymmetry and excess kurtosis. Therefore we need a model that can capture this phenomenon. The model is the Variance Gamma model, abbreviated as VG. The assumption of normality in the Black-Sholes-Merton theory is unable to capture the heavy tail and the asymmetry that exists in the log returns asset. This form of density usually that is too high compared to normal density is known as excess kurtosis. The additional parameters in the VG process can overcome heavy tail, asymmetry and excess kurtosis. These parameters can control the kurtosis, asymmetry, and adjust the slope of the log asset density. The VG process can be obtained from two approaches. First, the VG process as a change in Brownian motion to Gamma time, abbreviated as VG1. Second, the VG process is obtained by the difference of the two Gamma processes, abbreviated as VG2. In this study we applied the VG model to predict the daily stock price of PT Bank Negara Indonesia (Persero) Tbk. The results obtained show that the VG model provides accurate daily stock price prediction results.